7.23.2 problem 2

Internal problem ID [588]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 5. Linear systems of differential equations. Section 5.2 (Applications). Problems at page 345
Problem number : 2
Date solved : Saturday, March 29, 2025 at 04:57:23 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )-2 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=2 x \left (t \right )-3 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.111 (sec). Leaf size: 34

ode:=[diff(x(t),t) = x(t)-2*y(t), diff(y(t),t) = 2*x(t)-3*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-t} \left (c_2 t +c_1 \right ) \\ y \left (t \right ) &= \frac {{\mathrm e}^{-t} \left (2 c_2 t +2 c_1 -c_2 \right )}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 46

ode={D[x[t],t]==x[t]-2*y[t],D[y[t],t]==2*x[t]-3*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to e^{-t} (2 c_1 t-2 c_2 t+c_1) \\ y(t)\to e^{-t} (2 (c_1-c_2) t+c_2) \\ \end{align*}

✓ Sympy. Time used: 0.088 (sec). Leaf size: 36

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-x(t) + 2*y(t) + Derivative(x(t), t),0),Eq(-2*x(t) + 3*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = 2 C_{1} t e^{- t} + \left (C_{1} + 2 C_{2}\right ) e^{- t}, \ y{\left (t \right )} = 2 C_{1} t e^{- t} + 2 C_{2} e^{- t}\right ] \]